- Email: [email protected]
Electronic structure of (In,Ga) As(Ga, Al) As strained-layer quantum wells
58
Materials Science and Engineering, B20 (1993) 58-61
Electronic structure of (In, Ga)As-( Ga, A1 )As strained-layer quantum wells David J. Dunstan Department of Physics, Universityof Surre~, Guildford, Surrey GU2 5XH (UK)
Bernard Gil* Groupe d'Etude de Semiconducteurs, Universitdde Montpellier II, CNRS URA 357, Place Eugdne Batailton, 34095 Montpetlier Cedex 5 (France)
Abstract We discuss the growth of epitaxial structures of (In,Ga)As on GaAs substrates, with elastic and plastic strain, and study the effect of the corresponding built-in strain on the photoluminescence of quantum wells inserted intosuch buffer layers. In addition, we examine the effect of built-in strain on the full electronic structure of strained-layer quantum wells. Reflectivity data taken at 2 K, where we could measure ground state type t, as well as excited type I and type it transitions, have revealed that a correct description of these heterostructures requires including the effect of the spin-orbit split-off states in the valence band physics as soon as the configuration of the electron-to-light-hole potential profiles switches from type II to type I. We anticipate that this effect should be included for the calculation of threshold currents in quantum well lasers made from such heterostructures.
I. Introduction
2. Metallurgy and sample morphologies
Strained-layer semiconductor heterostructures are currently of great interest. These structures are stimulating both from the point of view of the device designer, through their potential for developing a new generation of solid state lasers, for instance, and from the point of view the physicist, owing to the presence of new physical phenomena which need to be understood. Most of the semiconductors studied are lattice mismatched to low cost, commercial substrates such as silicon or gallium arsenide. Therefore, it is necessary' to understand and characterize the effects of such lattice mismatches on the properties of devices based on such heterostructures. This paper is divided into two main parts. In the first part, we detail how to proceed to optimize the growth of strained-layer devices, with the lowest density lattice imperfections, on lattice-mismatched substrates. In the second part, we show that, in these heterostructures, the description of the valence band physics which has been invoked to affect the threshold current of lasers should not be restricted to the contribution of the F 8related states but requires the inclusion of the contribution of the states derived from the deep F 7 spin-orbit split-off state.
Firstly, we have grown both strained and relaxed layers of (In, Ga)As on GaAs by molecular beam epitaxy at substrate temperatures from 400 to 550 °C and at growth rates of 0.25 and 2 monolayers s- t. In growing thick (3/~m) relaxed buffer layers of InxGa ~_xAs on GaAs substrates, we find that there is a distinct qualitative difference in the layers grown at x--0.2 and below, and at x=0.3 and above. In the layers with low indium contents, the density of threading dislocations is low"and the surface morphology is at worst hatched, showing striations or grooving in the 110 directions. In the layers with higher indium contents, i.e. x = 0.3 and above, the density of threading dislocations is very high and the surface morphology is very bad, We have grown quantum wells into the top of such layers and, in the samples with high indium contents, no photoluminescence is observed from the quantum well, while, in those with low indium contents, the quantum wells luminesce quite brightly, with line widths in the region of 20 meV. These results are understandable within the framework of the concepts of two- and three-dimensional growth [1]. For small lattice mismatches, the pseudomorphic growth of a layer is not perturbed by the strain nor by the relaxation, while the dislocation densities and surface morphologies are the results of the relaxation rather than of the growth. At large lattice mis-
*Author to whom correspondence should be addressed. 0921-5107/93/$6.00
© 1993 - Elsevier Sequoia, All rights reserved
D. J. Dunstan, B. Gil
/
(In, Ga)As-(Ga, Al)As strained layer quantum wells
matches, the growth itself is perturbed, even at very small layer thicknesses of a few monolayers. Except under some special growth conditions, islanding occurs, or other severe growth perturbation, and the layer is heavily defected, irrespective of the relaxation. This results in a very large defect density and defect species which are not eliminated by continued growth nor by superlattice filters. The transition from two-dimensional growth (for Ax = 0.2 or less) to three-dimensional growth (Ax = 0.3 or more) is expected to be a result of the strain and not of the composition. Consequently, it is appropriate to make small compositional steps or to grade the composition. However, this is not a complete solution to the problem of obtaining large lattice constant changes, because we have evidence that a succession of small relaxation steps progressively damages the underlying layers and this prejudices the crystalline quality of the whole structure. We have grown quantum wells in stepped composition buffers up to x = 0 . 4 and obtained photoluminescence intensities as good as from quantum wells grown in a single, x = 0.1 buffer (Fig. 1). The photoluminescence from the quantum wells in the lower composition layers is progressively degraded when compositions with higher x values are grown on top. Thus, in Fig. 1, the weak peak ' ~ ' at 1150 nm on trace (b) should be compared with the strong emission of trace (a): both peaks are from quantum wells in In0.2GaaAs layers but the weak peak in trace (b) has
la)
2000
2"
E
o
o
Lv
Lv
c
co
O_
800
I ,I, 1000
Wovelength
, I 1200
I 1/.00
i
i
I
i
i i
I
I
i
(rim)
Fig. 1. Photoluminescence spectra (a) of a quantum well grown in relaxed In0.zGa0 8As and (b) of quantum wells grown as probes in the various layers of a stepped composition relaxed buffer layer. The peak in spectrum (a) should be compared with the peak marked ' ~ ' in spectrum (b).
59
suffered from the growth of the In0.4Ga0.6As layer on top. We believe that this is due to continued plastic relaxation caused by the subsequent growth. In the spectrum of trace (b), the luminescence of a quantum well in the underlying GaAs is also seen at 990 nm. Without the relaxed layers on top, this well would normally have a line width of about 2 meV; the growth of the relaxed layer on top has broadened it to some 20 meV. This demonstrates the propagation of damage downwards, even into the substrate, owing to the presence of the strained quantum well.
3. Electronic structure of heavily strained quantum wells When a layer is grown above a critical thickness, it relaxes plastically. It does not relax completely and, even at thicknesses orders of magnitude greater than the critical thickness, there is still a residual strain of 10-3_ 10- 4. How the buffer relaxes at this stage is not fully established. In this paper, we shall limit our theoretical exploration of the electronic structure of such samples for layer thicknesses ranging below the critical thickness of (In, Ga)As on GaAs. We have studied a series of InxGa~_xAs quantum wells with constant thickness (10nm), sandwiched between Ga~ _yAl,.As barrier layers 50 nm thick. The quantum well energy levels were determined by differential spectroscopy. We used the wavelength modulation of the 2 K reflectivity. Results presented here were taken on a series of five samples. They were chosen so that the series allowed us to explore the effect of the built-in stress by changing the indium concentration and to study the effect of the depths of the potential wells (i.e. the aluminium composition), while keeping a constant indium mole fraction. An indium concentration of around 18% leads to a built-in stress of some 15 kbar in the ternary compound. Figure 2 displays some of the wavelength-modulated reflectivity spectra taken at 2 K. All the spectra show similar features: a strong, low energy derivative structure that will be attributed to e~hh,, followed by a weaker structure e~lh, and, at higher energy, the e2hh 2 excited transitions are observed. To describe the kinetic energy hamiltonian, we limit ourselves to a model in which the conduction and valence band physics are decoupled from each other. This approximation does not hold in small-gap materials such as InAs; the conduction band then has to be included explicitly in the model and the Liittinger parameters have to be renormalized accordingly [2]. Considering states at ki = 0 and since Kramer's degeneracy is kept, in this model the effective mass
60
D. Y. Dunstan, B. Gil
eihhl
)
(a):
x
=
0.I6
y
=
(ia):
x = O,'lfi
y = 0.045
(cj:
x = 0.1E
y = 0.075
(cO:
x = 0.20
y = 0.~85
/
(In, Ga)As-(Ga, AI)As slruined hlyer quamum we/A
O
40 .,...i
c D
_J
ellhl
e2hl~2
_d k. U
>FO'3 Z
Ga
Ld
In 1-x
As-Ga x
:L-y
ellh I
I
,
1.3o
,
1.34
I
I
,
1.38
A2
As
y
e2hl~2
,
1.42
I
L
I. 46
ENERGY (mY) Fig. 2. Low temperature, wavelength modulation of the reflectance of Ga~_~InxAs-Ga~_yAlyAssingle quantum wells.
ICh)
I¢,)
-(Yl-2y2)kf 0
0 -()q + 2ye)k~2
0
2(21/2)y2k: 2
(3 I
q/(r) = ~ F,,(r)(ba(r) 2(2J/2)y2kz2 - A o -- Yl
kf
(1) The effect of built-in stress on the energies of valence states can be written as [2]
I¢.)
I¢,)
- av(ex~ + eyy + ezz) -b(e=-Gx)
0
0
-av(exx + eyy +ezz ) +b(ezz -exx) -21/2b'(ezz - ex~)
0
[ H,**~ + I/,,~( z)d,,~ + S H ~ e l F ~ = Ek), (I
where a = 1, so is the hole band index, and F(r) is a two-component envelope function. The electronic wave function tit(r) reads
hamiltonian is given by [2]
Hki n
A second degree of approximation consists first ot' resolving the strain problem, still neglecting the offdiagonal terms in eqn. (1). A new potential profile, accounting for the stress-induced effect of the spin-orbit split-off states is found. Then, the light-hole confinement energies are computed in a one-band approximation [3]. Finally, a degenerate band formalism can be used to solve the valence band problem. To obtain the eigenenergies and the analytical expressions of the eigenfunctions in the strained-layer quantum wells, we adapt the model developed by Andreani et al. [41]to treat the Ffl problem for the present situation. The one-band treatment of the electron and heavyhole confinement energies is made using a 33:67 strained heavy-hole-to-electron band offset ratio 15I, while the calculations of the light-hole and split-off hole confinement energies are made in a two-band calculation, replacing k: by -ifi d/Oz in eqns. (1) and (2). The effective mass equation for the coupled valence bands is written as
ez
~b~(r) being the ath Bloch state of the valence band. The envelope function approximation emphasizes the need to ensure the continuity of each component F~ of the envelope function and the continuity of the vector [2]
0
(2)
where esj are the components of the strain field, and the deformation potentials av, av', b and b' are defined in detail in ref. 2. The simplest model, which is the most commonly used, consists of neglecting any kind of coupling between the valence states. The effect of the built-in strain on the valence band is then limited to the F8v topmost states. This is quite satisfactory for weakly strained compounds with a large spin-orbit interaction.
-21/2b'(e:z -e.~) . + err . + ezz) -av ' (e,x
(Yt + 27=) O/Oz
l - 2(2~/2)Y2 3/Oz
-2(2~/2)), 20/Oz F,(r)
Y, O/3z
F~o(r)
The calculations of the optical subband-to-subband transition energies are shown in Fig. 3 for indium concentrations of 16%, 18% and 20%, and for aluminium concentrations ranging from 0% up to 20%.
D. J. Dunstan, B. Gil
/
X = 0.20
X = 0.i8
X = 0.16
(ln, Ga)As-(Ga, Al)As strained layer quantum wells
The mixing between the Fs' and F7V Bloch states changes significantly the envelope function when the depth of the quantum well is varied. Keeping in mind that the well thickness also has a drastic effect on the subband mixing, it is clear that a correct calculation of threshold currents in strained-layer quantum well lasers requires that the valence band physics are treated exactly.
1450
L
etlh t o
t400
tO C
ethh i
elhh t
4~
1350
D D
-
ethh t
[_
1300
61
,,,1 .... , .... I,,, 5 10 t 5 20
0
5
t0
t5
20
Aluminum concentration
5
t0
15
D
4. Conclusions
20
We have studied the built-in strain in (In, Ga)As grown on GaAs. This was done by using quantum wells inserted at various depths in epilayers as probes of the strain. Next, we examined the electronic structure and the optical properties of (Ga, In)As-(Ga, A1)As strained-layer quantum wells. The experimental results have proved that the inclusion of the interaction with the split-off valence band states is necessary to account for the valence band physics in such samples.
Y (%)
Fig. 3. Comparison between the calculated and measured transi•tion energies in Ga~ xlnxAs-Ga l_,Al~As single quantum wells 10 nm wide. Three values for x (0.i6, 0.18 and 0.20) were investigated (see upper part of the figure), while the aluminium concentration was varied continuously in each case. Concerning the ellh ~ transition, three calculations are shown: the simplest model (dotted lines), the model of ref. 3 (dashed lines) and the two-band approach (full lines).
The corresponding experimental data are represented by open squares. The full square data correspond to an indium composition of 17.5%. At low aluminium concentrations, the band-to-band energies calculated by our more sophisticated twoband approach (full line) match the values obtained by the simplest calculations. The disagreement between these three models increases slightly with the indium concentration. However, when y is increased, the three approximations give completely different results and only the two-band model fits the experimental findings. Moreover, the stronger the strain is (i.e. the value of x), the larger are the differences between the calculations. This point can be explained simply by considering that, even if the strain is strong, the coupling with split-off states will affect significantly the energies of the lighthole states when they are put into a type-I situation, where the confinement energy (and thus k~) is strong.
Acknowledgments We acknowledge K. J. Moore for fruitful discussions about the physics of (In, Ga)As, L. K. Howard for sample growth and E Boring for some of the numerical calculations.
References l D.J. Dunstan, Semicond. Sci. Technol., 6 ( 1991 ) A71. 2 B. Gil, L. K. Howard, D. J. Dunstan, P. Boring and E Lefebvre, Phys. Rev. B, 45 (1992) 3906. 3 S. H. Pan, H. Shen, Z. Hang, F. H. Pollak, W. Zhuang, Q. Xu, A. P. Roth, R. A. Massut, C. Lacelle and D. Morris,' Phys. Rev. B, 38 (1988) 3375. 4 L. C. Andreani, A. Pasquarello and F. Bassani, Phys. Rev. B, 36 (1987) 5887. 5 K.J. Moore, Inst. Phys. Conf Ser., 12_?(1992) 187 and references therein.
Materials Science and Engineering, B20 (1993) 58-61
Electronic structure of (In, Ga)As-( Ga, A1 )As strained-layer quantum wells David J. Dunstan Department of Physics, Universityof Surre~, Guildford, Surrey GU2 5XH (UK)
Bernard Gil* Groupe d'Etude de Semiconducteurs, Universitdde Montpellier II, CNRS URA 357, Place Eugdne Batailton, 34095 Montpetlier Cedex 5 (France)
Abstract We discuss the growth of epitaxial structures of (In,Ga)As on GaAs substrates, with elastic and plastic strain, and study the effect of the corresponding built-in strain on the photoluminescence of quantum wells inserted intosuch buffer layers. In addition, we examine the effect of built-in strain on the full electronic structure of strained-layer quantum wells. Reflectivity data taken at 2 K, where we could measure ground state type t, as well as excited type I and type it transitions, have revealed that a correct description of these heterostructures requires including the effect of the spin-orbit split-off states in the valence band physics as soon as the configuration of the electron-to-light-hole potential profiles switches from type II to type I. We anticipate that this effect should be included for the calculation of threshold currents in quantum well lasers made from such heterostructures.
I. Introduction
2. Metallurgy and sample morphologies
Strained-layer semiconductor heterostructures are currently of great interest. These structures are stimulating both from the point of view of the device designer, through their potential for developing a new generation of solid state lasers, for instance, and from the point of view the physicist, owing to the presence of new physical phenomena which need to be understood. Most of the semiconductors studied are lattice mismatched to low cost, commercial substrates such as silicon or gallium arsenide. Therefore, it is necessary' to understand and characterize the effects of such lattice mismatches on the properties of devices based on such heterostructures. This paper is divided into two main parts. In the first part, we detail how to proceed to optimize the growth of strained-layer devices, with the lowest density lattice imperfections, on lattice-mismatched substrates. In the second part, we show that, in these heterostructures, the description of the valence band physics which has been invoked to affect the threshold current of lasers should not be restricted to the contribution of the F 8related states but requires the inclusion of the contribution of the states derived from the deep F 7 spin-orbit split-off state.
Firstly, we have grown both strained and relaxed layers of (In, Ga)As on GaAs by molecular beam epitaxy at substrate temperatures from 400 to 550 °C and at growth rates of 0.25 and 2 monolayers s- t. In growing thick (3/~m) relaxed buffer layers of InxGa ~_xAs on GaAs substrates, we find that there is a distinct qualitative difference in the layers grown at x--0.2 and below, and at x=0.3 and above. In the layers with low indium contents, the density of threading dislocations is low"and the surface morphology is at worst hatched, showing striations or grooving in the 110 directions. In the layers with higher indium contents, i.e. x = 0.3 and above, the density of threading dislocations is very high and the surface morphology is very bad, We have grown quantum wells into the top of such layers and, in the samples with high indium contents, no photoluminescence is observed from the quantum well, while, in those with low indium contents, the quantum wells luminesce quite brightly, with line widths in the region of 20 meV. These results are understandable within the framework of the concepts of two- and three-dimensional growth [1]. For small lattice mismatches, the pseudomorphic growth of a layer is not perturbed by the strain nor by the relaxation, while the dislocation densities and surface morphologies are the results of the relaxation rather than of the growth. At large lattice mis-
*Author to whom correspondence should be addressed. 0921-5107/93/$6.00
© 1993 - Elsevier Sequoia, All rights reserved
D. J. Dunstan, B. Gil
/
(In, Ga)As-(Ga, Al)As strained layer quantum wells
matches, the growth itself is perturbed, even at very small layer thicknesses of a few monolayers. Except under some special growth conditions, islanding occurs, or other severe growth perturbation, and the layer is heavily defected, irrespective of the relaxation. This results in a very large defect density and defect species which are not eliminated by continued growth nor by superlattice filters. The transition from two-dimensional growth (for Ax = 0.2 or less) to three-dimensional growth (Ax = 0.3 or more) is expected to be a result of the strain and not of the composition. Consequently, it is appropriate to make small compositional steps or to grade the composition. However, this is not a complete solution to the problem of obtaining large lattice constant changes, because we have evidence that a succession of small relaxation steps progressively damages the underlying layers and this prejudices the crystalline quality of the whole structure. We have grown quantum wells in stepped composition buffers up to x = 0 . 4 and obtained photoluminescence intensities as good as from quantum wells grown in a single, x = 0.1 buffer (Fig. 1). The photoluminescence from the quantum wells in the lower composition layers is progressively degraded when compositions with higher x values are grown on top. Thus, in Fig. 1, the weak peak ' ~ ' at 1150 nm on trace (b) should be compared with the strong emission of trace (a): both peaks are from quantum wells in In0.2GaaAs layers but the weak peak in trace (b) has
la)
2000
2"
E
o
o
Lv
Lv
c
co
O_
800
I ,I, 1000
Wovelength
, I 1200
I 1/.00
i
i
I
i
i i
I
I
i
(rim)
Fig. 1. Photoluminescence spectra (a) of a quantum well grown in relaxed In0.zGa0 8As and (b) of quantum wells grown as probes in the various layers of a stepped composition relaxed buffer layer. The peak in spectrum (a) should be compared with the peak marked ' ~ ' in spectrum (b).
59
suffered from the growth of the In0.4Ga0.6As layer on top. We believe that this is due to continued plastic relaxation caused by the subsequent growth. In the spectrum of trace (b), the luminescence of a quantum well in the underlying GaAs is also seen at 990 nm. Without the relaxed layers on top, this well would normally have a line width of about 2 meV; the growth of the relaxed layer on top has broadened it to some 20 meV. This demonstrates the propagation of damage downwards, even into the substrate, owing to the presence of the strained quantum well.
3. Electronic structure of heavily strained quantum wells When a layer is grown above a critical thickness, it relaxes plastically. It does not relax completely and, even at thicknesses orders of magnitude greater than the critical thickness, there is still a residual strain of 10-3_ 10- 4. How the buffer relaxes at this stage is not fully established. In this paper, we shall limit our theoretical exploration of the electronic structure of such samples for layer thicknesses ranging below the critical thickness of (In, Ga)As on GaAs. We have studied a series of InxGa~_xAs quantum wells with constant thickness (10nm), sandwiched between Ga~ _yAl,.As barrier layers 50 nm thick. The quantum well energy levels were determined by differential spectroscopy. We used the wavelength modulation of the 2 K reflectivity. Results presented here were taken on a series of five samples. They were chosen so that the series allowed us to explore the effect of the built-in stress by changing the indium concentration and to study the effect of the depths of the potential wells (i.e. the aluminium composition), while keeping a constant indium mole fraction. An indium concentration of around 18% leads to a built-in stress of some 15 kbar in the ternary compound. Figure 2 displays some of the wavelength-modulated reflectivity spectra taken at 2 K. All the spectra show similar features: a strong, low energy derivative structure that will be attributed to e~hh,, followed by a weaker structure e~lh, and, at higher energy, the e2hh 2 excited transitions are observed. To describe the kinetic energy hamiltonian, we limit ourselves to a model in which the conduction and valence band physics are decoupled from each other. This approximation does not hold in small-gap materials such as InAs; the conduction band then has to be included explicitly in the model and the Liittinger parameters have to be renormalized accordingly [2]. Considering states at ki = 0 and since Kramer's degeneracy is kept, in this model the effective mass
60
D. Y. Dunstan, B. Gil
eihhl
)
(a):
x
=
0.I6
y
=
(ia):
x = O,'lfi
y = 0.045
(cj:
x = 0.1E
y = 0.075
(cO:
x = 0.20
y = 0.~85
/
(In, Ga)As-(Ga, AI)As slruined hlyer quamum we/A
O
40 .,...i
c D
_J
ellhl
e2hl~2
_d k. U
>FO'3 Z
Ga
Ld
In 1-x
As-Ga x
:L-y
ellh I
I
,
1.3o
,
1.34
I
I
,
1.38
A2
As
y
e2hl~2
,
1.42
I
L
I. 46
ENERGY (mY) Fig. 2. Low temperature, wavelength modulation of the reflectance of Ga~_~InxAs-Ga~_yAlyAssingle quantum wells.
ICh)
I¢,)
-(Yl-2y2)kf 0
0 -()q + 2ye)k~2
0
2(21/2)y2k: 2
(3 I
q/(r) = ~ F,,(r)(ba(r) 2(2J/2)y2kz2 - A o -- Yl
kf
(1) The effect of built-in stress on the energies of valence states can be written as [2]
I¢.)
I¢,)
- av(ex~ + eyy + ezz) -b(e=-Gx)
0
0
-av(exx + eyy +ezz ) +b(ezz -exx) -21/2b'(ezz - ex~)
0
[ H,**~ + I/,,~( z)d,,~ + S H ~ e l F ~ = Ek), (I
where a = 1, so is the hole band index, and F(r) is a two-component envelope function. The electronic wave function tit(r) reads
hamiltonian is given by [2]
Hki n
A second degree of approximation consists first ot' resolving the strain problem, still neglecting the offdiagonal terms in eqn. (1). A new potential profile, accounting for the stress-induced effect of the spin-orbit split-off states is found. Then, the light-hole confinement energies are computed in a one-band approximation [3]. Finally, a degenerate band formalism can be used to solve the valence band problem. To obtain the eigenenergies and the analytical expressions of the eigenfunctions in the strained-layer quantum wells, we adapt the model developed by Andreani et al. [41]to treat the Ffl problem for the present situation. The one-band treatment of the electron and heavyhole confinement energies is made using a 33:67 strained heavy-hole-to-electron band offset ratio 15I, while the calculations of the light-hole and split-off hole confinement energies are made in a two-band calculation, replacing k: by -ifi d/Oz in eqns. (1) and (2). The effective mass equation for the coupled valence bands is written as
ez
~b~(r) being the ath Bloch state of the valence band. The envelope function approximation emphasizes the need to ensure the continuity of each component F~ of the envelope function and the continuity of the vector [2]
0
(2)
where esj are the components of the strain field, and the deformation potentials av, av', b and b' are defined in detail in ref. 2. The simplest model, which is the most commonly used, consists of neglecting any kind of coupling between the valence states. The effect of the built-in strain on the valence band is then limited to the F8v topmost states. This is quite satisfactory for weakly strained compounds with a large spin-orbit interaction.
-21/2b'(e:z -e.~) . + err . + ezz) -av ' (e,x
(Yt + 27=) O/Oz
l - 2(2~/2)Y2 3/Oz
-2(2~/2)), 20/Oz F,(r)
Y, O/3z
F~o(r)
The calculations of the optical subband-to-subband transition energies are shown in Fig. 3 for indium concentrations of 16%, 18% and 20%, and for aluminium concentrations ranging from 0% up to 20%.
D. J. Dunstan, B. Gil
/
X = 0.20
X = 0.i8
X = 0.16
(ln, Ga)As-(Ga, Al)As strained layer quantum wells
The mixing between the Fs' and F7V Bloch states changes significantly the envelope function when the depth of the quantum well is varied. Keeping in mind that the well thickness also has a drastic effect on the subband mixing, it is clear that a correct calculation of threshold currents in strained-layer quantum well lasers requires that the valence band physics are treated exactly.
1450
L
etlh t o
t400
tO C
ethh i
elhh t
4~
1350
D D
-
ethh t
[_
1300
61
,,,1 .... , .... I,,, 5 10 t 5 20
0
5
t0
t5
20
Aluminum concentration
5
t0
15
D
4. Conclusions
20
We have studied the built-in strain in (In, Ga)As grown on GaAs. This was done by using quantum wells inserted at various depths in epilayers as probes of the strain. Next, we examined the electronic structure and the optical properties of (Ga, In)As-(Ga, A1)As strained-layer quantum wells. The experimental results have proved that the inclusion of the interaction with the split-off valence band states is necessary to account for the valence band physics in such samples.
Y (%)
Fig. 3. Comparison between the calculated and measured transi•tion energies in Ga~ xlnxAs-Ga l_,Al~As single quantum wells 10 nm wide. Three values for x (0.i6, 0.18 and 0.20) were investigated (see upper part of the figure), while the aluminium concentration was varied continuously in each case. Concerning the ellh ~ transition, three calculations are shown: the simplest model (dotted lines), the model of ref. 3 (dashed lines) and the two-band approach (full lines).
The corresponding experimental data are represented by open squares. The full square data correspond to an indium composition of 17.5%. At low aluminium concentrations, the band-to-band energies calculated by our more sophisticated twoband approach (full line) match the values obtained by the simplest calculations. The disagreement between these three models increases slightly with the indium concentration. However, when y is increased, the three approximations give completely different results and only the two-band model fits the experimental findings. Moreover, the stronger the strain is (i.e. the value of x), the larger are the differences between the calculations. This point can be explained simply by considering that, even if the strain is strong, the coupling with split-off states will affect significantly the energies of the lighthole states when they are put into a type-I situation, where the confinement energy (and thus k~) is strong.
Acknowledgments We acknowledge K. J. Moore for fruitful discussions about the physics of (In, Ga)As, L. K. Howard for sample growth and E Boring for some of the numerical calculations.
References l D.J. Dunstan, Semicond. Sci. Technol., 6 ( 1991 ) A71. 2 B. Gil, L. K. Howard, D. J. Dunstan, P. Boring and E Lefebvre, Phys. Rev. B, 45 (1992) 3906. 3 S. H. Pan, H. Shen, Z. Hang, F. H. Pollak, W. Zhuang, Q. Xu, A. P. Roth, R. A. Massut, C. Lacelle and D. Morris,' Phys. Rev. B, 38 (1988) 3375. 4 L. C. Andreani, A. Pasquarello and F. Bassani, Phys. Rev. B, 36 (1987) 5887. 5 K.J. Moore, Inst. Phys. Conf Ser., 12_?(1992) 187 and references therein.